The general purpose of an alignment calibration of a scanning lidar is to derive a function that allows for the transformation of coordinates from the internal coordinate system of the lidar into an external orthogonal coordinate system. The transformation matrix must be known in order to accurately target the correct measurement point and to determine the location for each measurement. The desired measurement points are usually defined in an external orthogonal coordinate system aligned with the vertical, south–north and west–east axes. The difference between the coordinate systems arises from multiple sources. Firstly, the lidar device is often not perfectly aligned with true north and may not be levelled during its setup, resulting in a tilt. Secondly, the scanning mechanism may exhibit deviations in the vertical lidar axis. This so-called elevation offset is independent of the azimuth angle and causes the laser beam to exit consistently higher or lower in all directions e.g.. Possible causes include misalignment of the mirrors within the scan head, as well as errors in the scan head's positioning, which may result from issues with the motor encoder or with the hall effect sensor, mechanical misalignment, motion control software, or related software parameters. These factors are important to ensure accurate measurements and reliable data from scanning lidar measurements.

The measurement location of a scanning lidar in the internal coordinate system is usually defined by the elevation angle φL, the azimuth angle θL, and the measurement range from the lidar r. Herein, the azimuth angle is defined clockwise from the y axis, with the device north corresponding to θL=0°. The elevation angle φL is defined relative to the internal horizontal plane, which is covered by the device's internal north and east vectors. Negative angles mean that the laser beam is pointing downward (see Fig. a). The range r from the lidar is defined by the time of flight of the laser pulse. The returned signal corresponds to a probe volume with a length that is determined by the length of the emitted pulse and the window used for signal processing. For the derivation of the method presented in this paper – which is based on the measured distance to the sea surface – it is convenient to approximate the probe volume by a single point and, thus, a single measurement range. The motivation for this approximation and its implications are discussed in detail in Sect. , and a sensitivity analysis is presented in Sect. . The following equation defines the relationship between φL, θL, r, and the lidar measurement location (xL): 1xL=r⋅cos⁡(φL)⋅sin⁡(θL)cos⁡(φL)⋅cos⁡(θL)sin⁡(φL). The device's tilt is characterised by pitch α and roll β. Note that the definitions of pitch and roll vary across the literature. Herein, pitch is defined as the rotation around the internal west–east axis of the device, whereby a positive pitch corresponds to a downward tilt in the direction of the device's north. Roll refers to rotation around the internal north–south axis, with a positive roll indicating a downward tilt toward the device's west (see Fig. a). Mathematically, the tilting by pitch and roll can be expressed by the following rotation matrices e.g.: 2Rpitch=1000cos⁡(α)sin⁡(α)0-sin⁡(α)cos⁡(α),3Rroll=cos⁡(β)0-sin⁡(β)010sin⁡(β)0cos⁡(β). The rotation matrices are defined to describe the transformation of the tilted lidar coordinate system into the external orthogonal coordinate system. Figure b illustrates this rotation in two dimensions, using pitch as an example. The multiplication of the two matrices results in a matrix that describes the inclination of the lidar device in terms of pitch and roll: 4Rtilt=Rpitch⋅Rroll=cos⁡(β)0-sin⁡(β)sin⁡(α)⋅sin⁡(β)cos⁡(α)sin⁡(α)⋅cos⁡(β)cos⁡(α)⋅sin⁡(β)-sin⁡(α)cos⁡(α)⋅cos⁡(β). Note that the order of rotations matters; thus, Rpitch⋅Rroll≠Rroll⋅Rpitch. The latter could also be a valid definition for Rtilt. However, in the context of scanning lidar devices, pitch and roll are usually small enough such that equality between both possible definitions can be assumed. In fact, if the absolute values of pitch and roll are both below 1°, the two definitions lead to a difference of, at most, 3.05 m of horizontal deviation and less than 0.03 m of vertical deviation at a measurement point within 10 km distance.

To reach a specific point in the external coordinate system, defined by the external elevation angle φE, the corresponding internal elevation angle φL must be determined. This requires accounting for the tilt of the device as the internal and external coordinate systems are misaligned due to pitch and roll (see Fig. b). However, besides tilt effects, vertical-axis deviations of the scanning mechanism (elevation offset) can also be relevant for alignment calibration. For the lidar considered here, Vaisala's WLS400S, a constant elevation offset Φ often occurs for technical reasons in the alignment of a scanner head (illustrated in Fig. a). The internal elevation angle φL is composed of the programmable angle φL′ and an elevation offset Φ: 5φL=φL′+Φ. Here, φL′ is the angle defined within the lidar system, while φL denotes the actual beam elevation, shifted by the offset and additionally affected by device tilt. A positive Φ corresponds to an upward shift relative to the ideal, offset-free orientation.

By applying Eqs. (), (), and (), the lidar's internal direction vector xL can be transformed into the corresponding external direction vector xE. This transformation is performed using a rotation matrix, assuming the coordinate origin is located at the laser beam exit point. 6xE=Rtilt⋅xL=r⋅cos⁡(β)⋅cos⁡(φL′+Φ)⋅sin⁡(θL)-sin⁡(β)⋅sin⁡(φL′+Φ)sin⁡(α)⋅sin⁡(β)⋅cos⁡(φL′+Φ)⋅sin⁡(θL)+cos⁡(α)⋅cos⁡(φL′+Φ)⋅cos⁡(θL)+sin⁡(α)⋅cos⁡(β)⋅sin⁡(φL′+Φ)cos⁡(α)⋅sin⁡(β)⋅cos⁡(φL′+Φ)⋅sin⁡(θL)-sin⁡(α)⋅cos⁡(φL′+Φ)⋅cos⁡(θL)+cos⁡(α)⋅cos⁡(β)⋅sin⁡(φL′+Φ) The azimuth angle θL is corrected by applying a north offset γ, which accounts for the rotational misalignment of the lidar relative to true geographic north (see Fig. a): 7θE=θL+γ. θE denotes the external azimuth angle in the orthogonal external coordinate system, and γ is defined as positive when the lidar is rotated clockwise relative to true north. Assuming small pitch, roll, and elevation angles, θL≈θL′, where θL′ represents the programmable internal azimuth angle set within the lidar system, excluding the mechanical offset in the azimuth direction. In this study, the north offset is assumed to be known from independent measurements (e.g. CNR mapping of a hard target) and is therefore not considered in the following discussion.

For the SSL method presented herein, the third component (z component) of Eq. () is evaluated to derive α, β, and Φ. The vertical (z-) component in the external coordinate system is given by the sine of the external elevation angle φE: 8sin⁡(φE)=zr. From this, the following relationship is obtained: 9sin⁡(φE)=cos⁡(α)⋅sin⁡(β)⋅cos⁡(φL′+Φ)⋅sin⁡(θL)-sin⁡(α)⋅cos⁡(φL′+Φ)⋅cos⁡(θL)+cos⁡(α)⋅cos⁡(β)⋅sin⁡(φL′+Φ).

SSL is a method for verifying the beam pointing of a scanning lidar in offshore or nearshore environments. In this sense, it can also be regarded to be a practical, in situ calibration of the laser beam alignment. For this purpose, the sea surface is used as a target. In its original form, as proposed by , the sea surface is considered to be a flat plane around the measuring device. If the laser beam of the instrument is directed toward the water surface, the distance from which the laser beam hits the water surface will remain the same in all directions at a constant negative elevation angle in the case of ideal levelling. Deviations from this behaviour are caused by a tilt of the device and, thus, α and/or β deviating from zero. Waves may affect the uncertainty of the results, as discussed in Sect. .

On this basis, introduced the SSL method based on PPI scans. The values of pitch and roll were determined based on the directional differences in the distances between the lens of the laser beam and the entry point into the water surface at a fixed negative elevation angle. In addition, and introduced an alternative procedure for applying the SSL method using RHI scans. In contrast to the PPI method, this technique does not analyse ranges at a fixed elevation angle. Instead, it determines the elevation angle for each azimuth direction at which a constant given distance to the water surface is reached. Figure  shows the two scan patterns schematically.

Both methods rely, in principle, on the relationship established in Eq. (), utilising the fact that 10sin⁡(φE)=-hlidarrw, where hlidar denotes the height of the lidar above the sea surface, and rw defines the distance to the sea surface where the laser beam dips into the water. As the elevation angle is defined as negative for downward directions, hlidar is treated as a positive quantity, and the negative sign in Eq. () ensures that φE correctly represents a negative (downward) angle.

At distances of approximately 1.5 km, it is important to take into account that the sea surface is not perfectly flat but, instead, follows the curvature of the Earth. For instance, over a horizontal distance d of 1.5 km, this curvature causes a vertical drop of roughly 0.2 m relative to a horizontal reference plane. At d=2.5 km, the drop increases to 0.5 m, and, at d=4 km, it reaches 1.26 m. This effect can be approximated by accounting for the height difference zearth due to Earth's curvature between the lidar position at (d=0) and a target point at a horizontal distance of (d=x) p. 456–459: 11hlidar,d=x=hlidar,d=0-zearth,d=x=hlidar,d=0-x22⋅R, where R denotes the Earth's radius (here, R=6371 km), and x is the horizontal distance from the lidar. Here, hlidar,d=0 represents the lidar height at the position of the lidar location d=0. For simplicity, we will denote hlidar,d=0 as hlidar in the following. If x≫hlidar, the path length can be approximated by the horizontal distance, i.e. x≈r. For distances greater than 1 km, the difference between the actual path length and the horizontal distance becomes negligible, affecting the curvature-induced height correction only at the third decimal place. Inserting Eq. () into Eqs. () and () and applying the approximation, the resulting expression represents the extended SSL model including the effects of Earth's curvature and Φ: 12-hlidar-rw22⋅Rrw=cos⁡(α)⋅sin⁡(β)⋅cos⁡(φL′+Φ)⋅sin⁡(θL)-sin⁡(α)⋅cos⁡(φL′+Φ)⋅cos⁡(θL)+cos⁡(α)⋅cos⁡(β)⋅sin⁡(φL′+Φ). By acquiring measurements over multiple azimuth θL and elevation φL′ angles in combination with the corresponding measurement distance rw, the system parameters hlidar, α, β, and Φ can be inferred from Eq. (). A suitable approach for this estimation is, for example, the application of a least-squares fitting method. use PPI scans for this purpose and simplify Eq. () by assuming Φ=0 and neglecting zearth. In contrast, our approach enables direct estimation of Φ within the fitting process, removing the need for such assumptions.

Making several approximations for small angles, Eq. () can be simplified. In most practical applications, α and β do not exceed 0.2°. Thus, cos⁡(α)≈cos⁡(β)≈1, sin⁡(α)≈α, and sin⁡(β)≈β provide a highly accurate approximation. Scanning the sea surface at larger distances, φL′ becomes small, and Φ is generally small for properly manufactured scanning lidars. Applying the approximations sin⁡(φL′+Φ)≈φL′+Φ and cos⁡(φL′+Φ)≈1 and rearranging Eq. () for φL′, the resulting expression represents a simplified equation of the extended SSL method: 13φL′=α⋅cos⁡(θL)-β⋅sin⁡(θL)-hlidar-rw22⋅Rrw-Φ. The functional form of Eq. () corresponds to the established sinusoidal representation of elevation angle deviations (φE-φL) as a function of azimuth, commonly used in traditional hard-target calibration .

The accuracy of this simplification is verified by comparing Eq. () with the simplified Eq. (). Even in extreme cases, with α, β, and Φ up to 1°, the deviations at φ=-3.5° are less than 0.01°. Under typical conditions, with α,β,Φ≤0.2°, the deviation at φ=-3.5° is less than 0.003° and is, therefore, negligible. For steeper elevation angles, for example, at φ=-7°, the deviation is less than 0.02° but increases to about 0.06° at φ=-10°. This demonstrates that the simplified formulation (Eq. ) is very accurate for small elevation angles, but its validity is limited for larger elevation angles. In this case, Eq. () should be applied.

In general, the extended SSL method is not limited to a specific scan type such as PPI or RHI. It is important that distance measurements to the water surface are available for different azimuth and elevation angles. With multiple RHI scans, one automatically obtains measurements for various elevation angles. In the case of PPI, at least two different elevation angles must be considered. The difference is that, in RHI scanning, the beam is continuously moved through elevation angles, and the returned signal is an integration between two elevation steps. Therefore, a small angular resolution should be chosen to achieve high accuracy in the determination of the elevation offset. Because the beam integrates over a finite elevation interval, the returned signal contains contributions from multiple ranges. However, since Eq. () is optimised with respect to the elevation angle in degrees, the resulting error is limited by the elevation angle step size used during scanning but is likely to be much smaller. In contrast, PPI scanning is performed at discrete elevation angles, while the laser beam is continuously moved in azimuth. This azimuthal integration has a very small influence on φE and, thus, the distance determination as long as azimuth intervals are chosen to be small enough (and as long as pitch and roll angles are small).

The SSL method requires an accurate estimation of the range rw between the laser emission point of a scanning lidar and the water surface at a fixed elevation and azimuth angle. For SSL, rw is derived from the evolution of CNR over the range. If the laser beam of a lidar hits a solid object, in general, a very strong back-scatter signal is obtained. However, in contrast, when the beam hits the sea surface, the signal usually exhibits a sharp decrease in CNR values around this distance . This behaviour, seen in Fig. , can be attributed to the ability of the water surface to absorb parts of the laser light and/or reflect it in directions other than back to the lidar device.

suggested using the inflection point of the sharp drop-off of the (logarithmic) CNR signal over distance for determining the immersion point of the laser beam reaching the water, as seen in Fig. . For this purpose, they fitted an inverse sigmoid function to the observed CNR signal. The inflection point of this function is then considered to be the point at which the laser beam enters the water surface. A shortcoming of their method is that, at larger distances (flat elevation angles), the typical decrease in CNR values within the atmosphere for an increasing distance is not taken into account. suggested accounting for this effect by introducing a linear term (1+a⋅(r-ip)) into the sigmoid function. The modified equation for determining the distance, used herein, is 14CNR(r)=(CNRhigh-CNRlow)⋅(1+a⋅(r-ip))1+exp⁡((r-ip)⋅g)+CNRlow, where CNRhigh and CNRlow are the maximum and minimum values of the inverse sigmoid function, a is a constant that describes the shape of the CNR signal over a distance before the beam hits the water, and g is the growth rate of the function that describes the rate of decrease in the signal strength around the immersion point ip. Note that, due to the additional term, the inflection point is not exactly equal to ip. However, the difference is very small, and, for the further discussion, we will use the term inflection point interchangeably in relation to ip. The parameters a, ip, g, CNRhigh and CNRlow are derived by fitting Eq. () to the observations of the CNR signal data. In this analysis, the limits for a were defined as -0.01m-1≤a≤0m-1. These limits were derived from inspection of the derived ranges to exclude outliers and are not physically motivated. The parameter g was constrained to lie between 0 and 1 and was later filtered according to the criteria described in Sect. . Additionally, limits for ip, as well as the CNRhigh and CNRlow, were applied depending on the specific settings of the RHI or PPI scans.

While the inflection point provides a visually intuitive location for the distance to the water surface, its location does not directly correspond to the distance between the lidar and the water surface. Pulsed lidars average the signal over a certain distance – often called probe length – that is defined by the length and shape of the laser pulse and the internal signal processing e.g. of the lidar. When probing the atmosphere, the distance measure provided by the lidar should ideally correspond to the distance where approximately half of the signal returns from before that distance and half of the signal returns from behind that distance. If the weighting function along the beam of the lidar is assumed to be symmetric, this corresponds to the middle of the probe volume.

In contrast, due to the logarithmic scale of the CNR, the inflection point of the steep CNR decline corresponds to a distance with a signal several orders of magnitude smaller than at ranges before entering the water. This means that almost all of the probe volume is inside the water at the distance of the location of the inflection point. As a consequence, the distance of the inflection point needs to be corrected (reduced) to match the distance to the water surface.

In theory, the distance of the inflection point could be modelled by integrating the weighting function of the probe volume along the beam. The signal strength with the presence of a water surface (P^) at the distance bw as a function of the signal strength without the presence of a water surface (P) can be written as follows e.g.: 15P^(r)=∫-inf⁡bwχ(s+r)P(r)ds, where χ(s) is the weighting function of the signal along the beam. The exact modelling of the CNR drop-off with increasing r requires detailed knowledge of χ(s). While several studies have used models to describe the weighting inside the probe volume to, for example, model turbulence attenuation , the exact shape of χ(s) remains uncertain and depends on both pulse shape and signal processing and therefore also on the device type e.g.. Detailed information about these aspects was not available to the authors. Moreover, at the inflection point, as well for the majority of the steep CNR drop-off, the signal is several orders of magnitude smaller than the signal before the probe volume enters the water (see Fig. ). This makes modelling of the CNR drop-off extremely sensitive to assumptions about the tail end of χ(s) – an area where models of χ(s) are likely to be very uncertain.

For the reasons outlined above, rather than modelling the return signal as a function of range, a simple constant distance correction is applied in this paper. In the presented results, half the probe length rw=ip-37.5 m is chosen (probe volume length: 75 m). This choice is motivated by applying several simple assumptions about the change in the return signal with r. Firstly, we assume that r as provided by the lidar corresponds to the middle of the probe volume. Secondly, it is assumed that, outside of the probe volume length provided by the manufacturer, the share of the return signal has been reduced by several orders of magnitude (to the inflection point of the CNR-over-range curve), and (almost) no signal is returned from the distances closer than the water surface. We acknowledge that this is only a relatively rough approximation and therefore perform a sensitivity analysis in relation to ranging uncertainties in Sect. .

An important assumption that is made in Eqs. () and () is that the sea surface can be approximated by a flat surface. However, due to the presence of surface waves, this assumption is never fully satisfied in real-world conditions. Consequently, it is commonly advised to perform an SSL under calm wind conditions with minimal wave activity . However, the sea surface will never be entirely flat. In this analysis, we therefore assess the errors and/or uncertainties introduced into SSL by waves of different amplitudes and wavelengths. The characteristics of water waves depend on a variety of physical factors, e.g. wind speed, water depth, and currents . It is not possible to cover all conditions completely here. Therefore, a simplified and idealised simulation approach is chosen to quantify the influence of basic wave parameters on the measurement uncertainty of SSL.

While the SSL equation assumes that the surface is influenced exclusively by the curvature of the Earth (see Sect. ), a wave-shaped altitude profile of the sea surface is now introduced. This model considers a sinusoidal wave with varying amplitudes A, propagating along the x direction and characterised by a defined wavelength λ. For the analysis of wave effects, note that the x direction is arbitrary and does not affect the results because the lidar scan covers the full 360° circle. The water surface is divided into segments of length λ, with each segment i being assigned an amplitude Ai. The water surface is divided into segments of length λ, with each segment i being assigned an amplitude Ai. The resulting surface height zsur is obtained from the superposition of the wave and the curvature term of the Earth: 16zsur(x,y)=Ai⋅sin⁡2πλ(x+x0)-x2+y22⋅R. Here, x0=v⋅t describes a displacement of the wave along the x direction, where t is time, and v is wave velocity. The wave velocity is determined according to the linear dispersion relation for surface gravity waves in a finite water depth p. 125: 17v=geλ2πtanh⁡2πzdλ, where ge is the gravitational acceleration, and zd is the water depth.

For the amplitudes Ai of the individual wave segments, a Rayleigh distribution is assumed, which is often used to describe the statistical distribution of wave heights in wind-driven seas p. 33–36. Assuming that the waves are predominantly regular, sinusoidal shapes with random superposition, a relationship between the scale parameter (σ) of the Rayleigh distribution and the significant wave height (Hs) can be approximated as Hs≈4⋅σ . Here, the significant wave height Hs, defined as the arithmetic mean of the upper third of wave heights within an observation period, provides a statistical measure that directly determines the scale parameter σ of the Rayleigh distribution. In the simulation, specific amplitudes Ai are drawn for each wavelength segment i from the Rayleigh distribution with the scale parameter σ.

The amplitude distribution remains constant across the entire domain, while the wave shifts over time via the phase parameter x0. This generates realistic, non-stationary waves with different amplitudes, which are shifted in relation to the position of the lidar. Figure a and b show examples of the simulated wave model for selected elevation and azimuth angles and a specific significant wave height. The first intersection point between the laser beam and the wave is determined (red points), and the distance is calculated. A wave movement is simulated by changing x0, accounting for the time required by the lidar to change between beam directions.

The experimental results in this study are based on an offshore measurement campaign conducted in the North Sea using a WindCube WLS400S (Vaisala) long-range scanning lidar installed on a transition piece of a wind turbine (see Fig. a). Figure b illustrates the measurement setup of the wind farm, indicating the location of the scanning lidar at the southern edge of the wind farm. The circles represent example distances, showing that other wind turbines (blue dots) within the farm fall within the laser beam's line of sight at various ranges for specific azimuth angles. The SSL measurements were carried out during periods with wind speeds of <4 ms-1, from 19 November 2023 at 22:00:00 UTC to 20 November 2023 at 08:00:00 UTC, ensuring that the wind turbine was not running. This minimises the effects of platform inclination caused by the thrust of the wind turbine during the measurement. For this date, the sea state charts from the Federal Maritime and Hydrographic Agency (BSH) indicate significant wave heights of up to 1 m . These wave heights correspond to a wavelength of approximately 25 m, which corresponds to a typical wave steepness of 0.04 for deep water . The water depth during this measurement campaign was approximately 40 m.

The settings used for the measurements were as follows (see Table ): two sequences of SSLs were carried out using the RHI method (see Fig. a). The RHI scans of the first sequence (RHI 1) ranged from a start elevation angle of φstart -1.5° to an end elevation angle of φend -0.3°, and the second sequence (RHI 2) ranged from -3 to -1.5°, both with a step size of 0.02°. For one RHI measurement routine, a sequence of RHI scans was performed at discrete azimuth angles between 180 and 45°, with an azimuthal step size of 5°. The laser beams with azimuth angles outside this range were blocked by the tower. This sequence was followed by PPI scans (see Fig. b) with elevation angles of -3 to -0.5° in 0.5° steps. The resolution in the azimuth angle was 1°. A full run of the RHI routine required approximately 30 min. The six PPI scans used to perform a PPI-based SSL took approximately 25 min. During the observation period, this allowed for the acquisition of seven to eight complete SSL datasets per measurement configuration.

All scans were performed with an accumulation time of 500 ms and a probe length of 75 m, which is the smallest probe volume of a WLS400S. The resolution of the distance was adjusted depending on the measurement setting and the maximum distance as only a maximum number of 319 range gates was possible. In order to eliminate any backlash effects, a single line-of-sight (LOS) scan located just below the measurement setting was performed before each PPI and RHI scan so that each scan was always initiated from the same direction. Due to the wind turbine tower blocking the line of sight, no measurements could be obtained in the azimuth range between 45 and 180°. For comparison, 10 min mean values from an inclinometer installed inside the wind turbine at the platform level were used. A hard-target calibration carried out by the external consultancy (DNV, personal communication, 2022) was used to validate the estimation of Φ.

To minimise the impact of inaccurate distance estimations from the CNR-over-range signal, automated and targeted data filtering was applied based on the following criteria:

Initial value filter. Measurements with an initial CNR value below -21 dB were excluded as it is assumed that the laser beam was already obstructed in the first few metres. In our measurement campaign, the PPI scans cover the entire azimuth range (0–360°), which means that the laser hits the wind turbine and, many measurements are filtered out. In contrast, only 2 %–3 % of the RHI measurements are affected, e.g. by the railing.

The accuracy of the SSL results for pitch α, roll β, and elevation offset Φ (Sect. ; Eq. ) strongly depends on the assumptions made about the range of the laser's entry point into the sea surface rw (Sect. ) and a good approximation of the sea surface, which is influenced by wave-induced effects. The following sections present a sensitivity analysis of these factors.

This section presents the experimental evaluation of the extended SSL method using data from the scanning lidar WLS400S (Sect. ), with data filtering as described in Sect. . The extended SSL method (model) is examined by comparing its predictions with measured data using the RMSE as an indicator of model agreement (Sect. ). Subsequently, the SSL performance is evaluated for different RHI and PPI scan settings by repeating measurements over a period of time and comparing them with external data for the elevation offset (Sect. ).